In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w（x） = /x/2αe-（x4＋tx2）, x ∈R, where α is a constant larger than - 1/2 and t...In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w（x） = /x/2αe-（x4＋tx2）, x ∈R, where α is a constant larger than - 1/2 and t is any real number. They consider this problem in three separate cases： （i） c 〉 -2, （ii） c = -2, and （iii） c 〈 -2, where c ：= tN-1/2 is a constant, N = n ＋ a and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μ is a single interval, whereas in the third case the support of μt consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou （1993）.展开更多
In this paper, we study the asymptotics of the Krawtchouk polynomials Kn^N（z;p,q） as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c E （0, 1） ...In this paper, we study the asymptotics of the Krawtchouk polynomials Kn^N（z;p,q） as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c E （0, 1） as n →∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal, Our method is based on the Riemann-Hilbert approach introduced by Delft and Zhou.展开更多
The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms...The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe?cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11771090,11571376)
文摘In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w（x） = /x/2αe-（x4＋tx2）, x ∈R, where α is a constant larger than - 1/2 and t is any real number. They consider this problem in three separate cases： （i） c 〉 -2, （ii） c = -2, and （iii） c 〈 -2, where c ：= tN-1/2 is a constant, N = n ＋ a and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μ is a single interval, whereas in the third case the support of μt consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou （1993）.
基金Project supported by the the Research Grants Council of the Hong Kong Special Administrative Region,China (No. CityU 102504).
文摘In this paper, we study the asymptotics of the Krawtchouk polynomials Kn^N（z;p,q） as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c E （0, 1） as n →∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal, Our method is based on the Riemann-Hilbert approach introduced by Delft and Zhou.
文摘The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe?cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.
